List of shapes with known packing constant

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The packing constant of a geometric body is the largest average density achieved by packing arrangements of congruent copies of the body. For most bodies the value of the packing constant is unknown.[1] The following is a list of bodies in Euclidean spaces whose packing constant is known.[1] Fejes Tóth proved that in the plane, a point symmetric body has a packing constant that is equal to its translative packing constant and its lattice packing constant.[2] Therefore, any such body for which the lattice packing constant was previously known, such as any ellipse, consequently has a known packing constant. In addition to these bodies, the packing constants of hyperspheres in 8 and 24 dimensions are almost exactly known.[3]

Image Description Dimension Packing constant Comments
Rhombic dodecahedra.png
 Monohedral prototiles all 1 Shapes such that congruent copies can form a tiling of space
2-d pentagon packing dual.svg
 Regular pentagon 2 [math]\displaystyle{ \frac{5 - \sqrt{5}}{3} \approx 0.92131 }[/math] Thomas Hales and Wöden Kusner[4]
Smoothed Octagon Packed.svg
 Smoothed octagon 2 [math]\displaystyle{ \eta_{so} = \frac{ 8-4\sqrt{2}-\ln{2} }{2\sqrt{2}-1} \approx 0.902414 }[/math] Reinhardt[5]
Regular decagon.svg
 All 2-fold symmetric convex polygons 2 Linear-time (in number of vertices) algorithm given by Mount and Ruth Silverman[6]
FCC closed packing tetrahedron (20).jpg
 Sphere 3 π/√18 ≈ 0.7404805 See Kepler conjecture
Red cylinder.svg
 Bi-infinite cylinder 3 π/√12 ≈ 0.906900 Bezdek and Kuperberg[7]
Half-infinite cylinder 3 π/√12 ≈ 0.906900 Wöden Kusner[8]
Small rhombicuboctahedron.png
100px|center		 All shapes contained in a rhombic dodecahedron whose inscribed sphere is contained in the shape 3 Fraction of the volume of the rhombic dodecahedron filled by the shape Corollary of Kepler conjecture. Examples pictured: rhombicuboctahedron and rhombic enneacontahedron.
Hypersphere 24 [math]\displaystyle{ \frac{(\frac {\pi}{2})^{12}}{12!} \approx 0.000000471087 }[/math] See Hypersphere packing

References

  1. ↑ 1.0 1.1 Bezdek, András; Kuperberg, Włodzimierz (2010). "Dense packing of space with various convex solids". arXiv:1008.2398v1 [math.MG].
  2. ↑ Fejes Tóth, László (1950). "Some packing and covering theorems". Acta Sci. Math. Szeged 12. 
  3. ↑ Cohn, Henry; Kumar, Abhinav (2009). "Optimality and uniqueness of the Leech lattice among lattices". Annals of Mathematics 170 (3): 1003–1050. doi:10.4007/annals.2009.170.1003. 
  4. ↑ Hales, Thomas; Kusner, Wöden (2016). "Packings of regular pentagons in the plane". arXiv:1602.07220 [math.MG].
  5. ↑ Reinhardt, Karl (1934). "Über die dichteste gitterförmige Lagerung kongruente Bereiche in der Ebene und eine besondere Art konvexer Kurven". Abh. Math. Sem. Univ. Hamburg 10: 216–230. doi:10.1007/bf02940676. 
  6. ↑ Mount, David M.; Silverman, Ruth (1990). "Packing and covering the plane with translates of a convex polygon". Journal of Algorithms 11 (4): 564–580. doi:10.1016/0196-6774(90)90010-C. 
  7. ↑ Bezdek, András; Kuperberg, Włodzimierz (1990). "Maximum density space packing with congruent circular cylinders of infinite length". Mathematika 37: 74–80. doi:10.1112/s0025579300012808. 
  8. ↑ Kusner, Wöden (2014). "Upper bounds on packing density for circular cylinders with high aspect ratio". Discrete & Computational Geometry 51 (4): 964–978. doi:10.1007/s00454-014-9593-6. 



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